Download Singular-phase nano-optics in plasmonic

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DOI: 10.1038/NMAT3537
Singular-phase
nano-optics in plasmonic metamaterials
Supplementary Information to “Singular‐phase nanooptics in plasmonic for label-free
single-molecule detection
metamaterials: towards label‐free single molecule detection” V. G. Kravets1, F. Schedin1, R. Jalil1, L. Britnell1, R. V. Gorbachev1, D. Ansell1, B. Thackray1, K. S. Novoselov1, A. K. Geim1, A. V. Kabashin2 and A. N. Grigorenko1 1
2
School of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, UK Laboratoire Lasers, Plasmas et Procédés Photoniques (LP3, UMR 7341 CNRS), Faculté des Sciences de Luminy, Aix‐Marseille University, 163 Avenue de Luminy, 13288 Marseille Cedex 09, France 1. Darkness and sharp phase changes. Let us consider the phase of the light reflected from an optical system in the complex polarization plane E, where the phase corresponds to the polar angle, see Fig. S1(a). In this representation the x‐axis corresponds to the in‐phase component of the electric field electric 
field E x Ein cos(t  kr ) while the y‐axis corresponds to the out‐phase component of the electric field 
E y Eout sin(t  kr ) . (a)
(b)
180
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Figure S1. Phase properties of light. (a) Phase as a polar angle in the complex polarization plane E. (b) 3D representation of the phase. The blue and purple curves show sharp phase changes when the reflection from the sample nears zero (|E|=0 ‐ the point of darkness represented by the white dot) which is shown by grey arrows. The dotted grey line corresponds to the Heaviside phase jump. The colourmap on the right represents phase. Being plotted as a function of incidence angle (or light wavelength), the reflected electric field E forms a family of curves in the complex plane. For simplicity, in Fig. S1 we have shown two straight lines (blue 1
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and purple), although the results obtained are general and valid for curves of an arbitrary shape1. When these curves come close to the coordinate origin (which corresponds to the point of darkness |E|=0) the angle (or wavelength) dependence of the reflected light’s phase undergoes a dramatic change. Between the purple curve with a monotonic decrease of phase and the blue curve with an increase of phase we find the dotted grey curve which corresponds to a 180‐jump in phase1. It is necessary to note that phases of neighbouring points in the family of curves are close to each other. It is the phase “topology” that changes phase behaviour. Indeed, the position of the coordinate origin is different with respect to the blue and purple curves of Fig. S1. In addition, the phase changes become much quicker as the curves get closer to the origin (the point of darkness), which is shown by the grey arrows in Fig. S1(a). In general, if light reflection from a system crosses zero, then, phase of light demonstrates a Heaviside‐like 180‐jump as well as markedly different behaviour for close values of some external parameters. For example, the phase of p‐polarized light reflected from an optical interface near the Brewster angle exhibits a 180‐jump2; the phase of light reflected from a Fabry–Perot interferometer demonstrates a 180‐jump when the reflected light amplitude crosses zero2; and phase of light near the focus of a lens demonstrates 180‐jumps along the lens axis at the points where the light intensity is zero2. There exists subtle similarity between the Aharonov‐Bohm effect, the Berry phase and the phase of light at conditions of topological darkness. In all three cases, the parameter space of the phase is not simply connected. For the Aharonov‐Bohm effect, the wavefunction (and its phase) is not defined at the position of the solenoid; for the Berry phase, the phase is not defined at some combination of external parameters (normally at zero magnitude of the parameter vector); for the case of topological darkness, the phase is not defined at the line of zero reflection. 2. Enhanced phase sensitivity near the points of darkness. The complete darkness and underfined phase cannot be directly used for metrology. It is easy to see, however, that the closer to the condition of complete darkness we can work, the higher the phase sensitivity of an optical system will be3,4. Indeed, let us consider some plasmonic system which is characterised by a resonance reflection curve and compare phase and amplitude sensitivity of the curve. We show that the improvement of phase sensitivity over amplitude one is governed by the value of the intensity of reflected light at the resonance minimum. We introduce a figure of merit, FOM, as a ratio of the phase over the amplitude sensitivity FOM   /  , where  and  are changes of the phase and amplitude ellipsometric parameters under a change of environment. Neglecting the change of reflection for s‐polarization, we can roughly evaluate the phase changes near the resonance minima as    E p ,min / E p ,min (here E p ,min is the reflected electric field at the resonance minimum and  E p ,min is a change of this field due to environment or reaction) and the amplitude changes at the slope of the 2
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resonance as    E p ,sl / E p ,sl (here E p ,sl is the reflected electric field at the slope of the resonance and  E p ,sl is a change of this field due to environment or reaction). Taking into account that normally, for deep plasmonic resonances,  E p ,min   E p , sl and E p ,sl  E s , sl  E s ,min (which represent the fact that reflection for p‐and s‐polarizations are comparable at the slope of the resonance and s‐polarized reflection does not change a lot for a resonance curve), we get FOM
=  /   E p ,sl / E p ,min  E s ,min / E p ,min . Using the fact that for deep resonance the ellipsometer parameter in the reflection minimum can be evaluated as  min  E p ,min / E s ,min we finally obtain the expression: FOM  1 /  min , where  min is expressed in radians. Thus, low values of the minimum of reflection at plasmonic resonance guarantee very high phase sensitivity compared to amplitude one. For example, for  min  0.3 (which can be easily achieved using diffractive coupled plasmon resonances) we have FOM~200. The discussion of the phase sensitivity and experimental verification of the formula FOM  1 /  min is given in4. 3. Improvement of sensitivity due to averaging and fitting procedures: comparison of sensitivities of “simple” transduction schemes with “non‐simple” schemes. In our work we discuss methods that guarantee high raw sensitivity of localised plasmon resonance (LPR) measurements in a “simple” optical transduction scheme. By a “simple” optical sensing scheme we imply a detection of changes of intensity or phase of the light reflected from/transmitted through a simple plasmonic system in case where angle of incidence and/or light wavelength are fixed. It is well known, that one can improve on this “simple” sensing approach by using various methods of averaging and fitting. These methods include averaging signals over many nanostructures, averaging a number of spectra acquired at different times, using a large number of spectral points with idea to fit reflection (transmission) spectra and find the position of the plasmonic resonance with better accuracy, etc. These procedures can result in a strong increase of sensitivity depending on the number of averaged spectra, the number of spectral points used for fitting procedure and the number of nanostructures involved. In principle, the improvement can be significant. Indeed, 3‐orders enhancement of sensitivity of “simple” amplitude Surface Plasmon Resonance (SPR) has been achieved by fitting the SPR resonance curve with the help of weighted centroid algorithm5 and 3‐orders of magnitude enhancement of sensitivity of “simple” amplitude LPR was reported6 with the help of averaging and fitting. In Ref. 6 three different methods have been used to achieve areal mass sensitivity 3
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~1pg/mm2 for amplitude LPR (at a recording time of several minutes): a large number of nanoholes (~106) have been interrogated simultaneously, about 50 different spectra have been temporally averaged and the fitting of the resonance spectra with 103 points have been additionally used. It is necessary to stress that the above‐stated fitting and averaging procedures are not related to "true" sensing transduction (characterizing physical, chemical or biochemical phenomena of interest) and can be applied as an additional tool for the treatment of signals in all sensing geometries and modalities (including the ones described in this manuscript). Hence, real sensitivity of a method should be clearly separated from the one that involves additional tools. To make things as simple and as clear as possible, in our manuscript we discuss only “simple” sensitivities which are not improved by various fitting procedures. The measured in graphene hydrogenation experiments areal mass phase sensitivity of ~10fg/mm2 (which represents the detection limit) is a “simple” one – the light phase at a given wavelength and angle of incidence is deduced as a function of a ratio of in‐phase over out‐of‐phase light intensities. Since Woollam ellipsometer does not guarantee an excellent phase stability, we also provide the recalculated phase sensitivity 100atto‐g/mm2 (based on the thermally stabilised system with advanced phase extraction which provides much better phase stability than those in the spectroscopic ellipsometer used in our measurements but still does not use any fitting or averaging procedures). 4. Diffractive coupled localised plasmon resonances. The underlying physics of diffractive coupled localised plasmon resonances (DCLPR) is simple. It is known that regular periodic structures can show abrupt changes in reflection, which are referred to as Wood anomalies7. For regular arrays of nanoparticles on a transparent substrate these anomalies have been explained by Rayleigh as a disappearance of a diffracted beam when it crosses the boundary between ambient medium (which we assume to be air for simplicity) and substrate. A transition of a diffraction mode between air and substrate is not allowed due to different dispersion relations for light in both media. As a result, the diffraction mode is cut off at the so‐called Rayleigh cut‐off wavelength8. There are two types of Rayleigh cut‐off wavelengths (R) – one for the disappearance of an “air” diffraction mode (where a diffraction mode crosses the sample boundary from air to substrate) and another for the disappearance of a “substrate” diffraction mode (where a diffraction mode crosses the boundary from substrate to air)8. For example, the “air” R for a square array of a lattice constant a are given by9 Rair_
mpq
a 2
p  q 2  sin( )( p cos( )  q sin( )) 

m
where m, p and q are integers,  is the angle of incidence (AOI) and  is the polar angle of the substrate orientation with respect to the plane of incidence. For  =0 the most important “air” and “substrate” R are given by9 4
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a
a
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(1  sin( )) and 
( ns  sin( )) , R_m
m
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_m
where m is an integer and ns is the substrate refractive index. When an array is illuminated by the light of the Rayleigh cut‐off wavelength, one of the diffracted waves travels exactly along the surface of the substrate (at a grazing angle) and hence interacts with many nanoparticles. Therefore, if R is close to the wavelength of the LPR for an individual nanoparticle one might hope to achieve strong transfer of energy from the incident beam into LP modes in a narrow wavelength range near a Wood anomaly and hence obtain sharp plasmon resonances. In this case the nanoparticles array generates a mixed mode of LPR vibrations combined with the diffracted grazing wave which will be resonantly excited by incident light. For deep plasmonic nanostructures the formulae above should be modified by introducing an effective refractive index for air and substrate (which will be affected by the presence of nanostructures.) These formulae normally give a very good estimate for wavelengths of DCLPR9. 5. Topological darkness in plasmonic nanostructures. We have experimentally found that structures with DCLPR shown in Fig. S2 demonstrate topological darkness (zero reflection at some wavelength  and angle of incidence ). Figure S2. Structures demonstrating zero reflection at DCLPR. Yellow colour represents gold, blue – glass, grey metallic colour shows optional Cr, purple colour represents PMMA. A strong dip in reflection (and zero reflection) was normally observed close to the Rayleigh cut‐off a (1  sin( )) , where a is the period of the structure. Figure S3 below wavelength at m=1, R
_1
air
provides the experimental ellipsometric reflection parameter  for these 5 structures. One can see a pronounced dip in reflection near ~600nm for all structures. Zero reflection |rp|=0 at some wavelength was achieved by adjusting angle of incidence. 5
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It is necessary to stress that our detection device was a photodiode array spectrophotometer and hence we measured digitized spectra with the wavelength step of ~1nm. Since DCLPR are extremely narrow (resonance half‐width could be just 3nm), it means that the zero reflection point could be missed during digitization of the spectrum. Figure S3(f) shows the zoomed region of DCLPR near 630nm for the structure of Fig. S3(e) and we can see that the red points (corresponding to the measured spectral data) miss the zero reflection point at 633nm. In this case we use the linear extrapolation (see the blue lines of Fig. S3(f)) to find the minimum value of reflection and the position of the topological darkness. Table 1 gives the values of the ellipsometric reflection parameter near the point of darkness. Resonance Value min position extrapolated d=118nm,  min=613nm a=320nm  min=71o min=0.05o d=100nm,  min=599nm s=140nm,  min=63o min=‐0.08o a=320nm Touching d=134nm,  min=568nm double dots s=140nm,  min=65o min=0.1o a=320nm b=150nm,  min=605nm a=320nm  min=71o min=0.1o d=120nm,  min=633nm s=140nm,  min=59o min=0.03o a=320nm Geometry Single dots Double dots Nanostripes Double holes Sizes Table 1. Parameters at which zero reflection was observed for different structures shown in Fig. S2. We can see that within experimental error (which was ~0.1) the reflection from all samples depicted in Fig. S2 is close to zero at the resonance conditions. More detailed analysis involving the phase of light confirms the fact that the reflection from a nanostructured plasmonic metamaterial can indeed show darkness and a 180 Heaviside‐like jump of light phase9. We found that the double dot arrays are the most versatile for achieving darkness as they provide more possibility to tune the localized plasmon resonances of the structure to the Rayleigh cut‐off conditions by tuning the separation in the nanodot pair. At the same time, DCLPR of the highest quality factor were observed in the double‐hole array shown in Fig. S3(e). 7
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To conclude this section, we note that digital nature of the ellipsometric spectra could lead to measurement problems connected with discreetness of the spectrum. Also, to perform sensing measurements we do need some amount of light in order to register light phase. Hence, we had to detune from the point of complete darkness to a point where phase changes could be measured by our apparatus. For this reason, the experimental reflection spectra shown in Fig. 3 and 4 of the main manuscript do not demonstrate zero reflection. At the same time, the closer to the topological darkness we can work, the higher phase sensitivity of our installation will be. This can be achieved by using more advanced photodiode array spectrophotometers and/or different diffraction gratings. 6. Control experiments for hydrogenation. For completeness, in this section we present typical SEM and AFM pictures of the graphene transferred on the plasmonic nanoarray and the control experiment which shows that the plasmonic nanoarray has not been affected by the hydrogenation procedure. Graphene flake Gold stripes Graphene flake Gold stripes Figure S4. SEM (top) and AFM (bottom) pictures of graphene transferred on a plasmonic nanoarray. 8
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Figure S4 shows that the transferred graphene flake is flat and in direct contact with a gold nanostructure. We found that when graphene was not in “Van der Waals” contact with gold, it was unstable and rolled in scrolls. Figure S5 shows the typical ellipsometric spectra (close to the point of darkness) of one of our structure before the graphene transfer and after the set of measurements connected with graphene hydrogenation and subsequent mild plasma etching of the graphene flake from the sample. One can see that the spectrum returned to its original shape. after graphene etch
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Figure S5. Ellipsometric parameters  and  for nano‐stripe sample before graphene transfer (red line) and after final graphene etch (black line). References. 1 2 3 4 5 6 7 8 9 Grigorenko, A. N., Nikitin, P. I. & Kabashin, A. V. Phase jumps and interferometric surface plasmon resonance imaging. Applied Physics Letters 75, 3917‐3919 (1999). Born, M. & Wolf, E. Principles of Optics. (Cambridge University Press, 1980). Kabashin, A. V., Patskovsky, S. & Grigorenko, A. N. Phase and amplitude sensitivities in surface plasmon resonance bio and chemical sensing. Opt. Express 17, 21191‐21204 (2009). Kravets, V. G., Schedin, F., Kabashin, A. V. & Grigorenko, A. N. Sensitivity of collective plasmon modes of gold nanoresonators to local environment. Opt. Lett. 35, 956‐958 (2010). Johansen, K., Stålberg, R., Lundström, I. & Liedberg, B. Surface plasmon resonance: instrumental resolution using photo diode arrays. Measurement Science and Technology 11, 1630 (2000). Dahlin, A. B., Tegenfeldt, J. O. & Höök, F. Improving the Instrumental Resolution of Sensors Based on Localized Surface Plasmon Resonance. Analytical Chemistry 78, 4416‐4423, doi:10.1021/ac0601967 (2006). Wood, R. W. XLII. On a remarkable case of uneven distribution of light in a diffraction grating spectrum. Philosophical Magazine Series 6 4, 396‐402, doi:10.1080/14786440209462857 (1902). Rayleigh, L. On the Dynamical Theory of Gratings. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 79, 399‐416 (1907). Kravets, V. G., Schedin, F. & Grigorenko, A. N. Extremely Narrow Plasmon Resonances Based on Diffraction Coupling of Localized Plasmons in Arrays of Metallic Nanoparticles. Physical Review Letters 101, 087403 (2008). 9
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